Chicago Journal of Theoretical Computer Science

Volume 2010

Article 3

Published by the Department of Computer Science, The University of Chicago.

$d$-collapsibility is NP-complete for $d \ge 4$

Martin Tancer
Department of Applied Mathematics and Institute for Theoretical Computer Science
Faculty of Mathematics and Physics, Charles University
Malostranské ná 25, 118 00 Prague, Czech Republic

June 21, 2010

A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most $d-1$ that is contained in a unique maximal face. We prove that the algorithmic question whether a given simplicial complex is $d$-collapsible is NP-complete for $d \ge 4$ and polynomial time solvable for $d\le 2$.

As an intermediate step, we prove that d-collapsibility can be recognized by the greedy algorithm for $d \le 2$, but the greedy algorithm does not work for $d \ge3$.

A simplicial complex is d-representable if it is the nerve of a collection of convex sets in ${\mathbb R}^d$. The main motivation for studying $d$-collapsible complexes is that every $d$-representable complex is $d$-collapsible. We also observe that known results imply that $d$-representability is NP-hard to decide for $d \ge 2$.

Submitted October 23, 2008, revised June 15, 2010, published June 21, 2010.

DOI: 10.4086/cjtcs.2010.003

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